Remark. Determining the minimum d-separation set has been well studied in causal inference [1]}. We leverage the algorithm of finding a minimum cost separator [2]} to identify \(\mathbf {W}\) . The discovery procedure usually requires the complete knowledge of the causal graph. However, in the situation where the d-separation set to be used as well as the associated conditional distributions \(P(\mathbf {z}|\mathbf {x}_{t,a})\)  are given, the remaining part of the algorithm will work just fine without the causal graph information. Moreover, the assumption of knowing \(P(\mathbf {z}|\mathbf {x_{t,a}})\)  follows recent research works on causal bandit. Generalizing the causal bandit framework to partially/completely unknown causal graph setting is a much more challenging but important task. A recent work [3]} tries to generalize causal bandit algorithm based on causal trees/forests structure.
